Abstract

We use the three-cornered-hat method to evaluate the absolute frequency stabilities of three different ultrastable reference cavities, one of which has a vibration-insensitive design that does not even require vibration isolation. An Nd:YAG laser and a diode laser are implemented as light sources. We observe ~ 1 Hz beat note linewidths between all three cavities. The measurement demonstrates that the vibration-insensitive cavity has a good frequency stability over the entire measurement time from 100 μs to 200 s. An absolute, correlation-removed Allan deviation of 1.4 × 10−15 at 1 s of this cavity is obtained, giving a frequency uncertainty of only 0.44 Hz.

© 2009 Optical Society of America

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  1. T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
    [CrossRef] [PubMed]
  2. A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  7. B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
    [CrossRef]
  8. A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]

2009

2008

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

2007

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007).
[CrossRef] [PubMed]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

2006

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83,531-536 (2006).
[CrossRef]

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

2005

T. Schneider, E. Peik, and C. Tamm, "Sub-Hertz optical frequency comparisons between two trapped 171Yb+ ions," Phys. Rev. Lett. 94,230801 (2005).
[CrossRef] [PubMed]

2004

2002

A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002).
[CrossRef]

1999

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
[CrossRef]

1993

A. Premoli and P. Tavella, "A revisited three-cornered hat method for estimating frequency standard instability," IEEE Trans. Instrum. Meas. 42,7-13 (1993).
[CrossRef]

1988

1983

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Alnis, J.

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

Becker, Th.

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Bergquist, J. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
[CrossRef]

Blatt, S.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007).
[CrossRef] [PubMed]

Boyd, M. M.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007).
[CrossRef] [PubMed]

Brusch, A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Campbell, G. K.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

Chen, L. S.

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

Chou, C. W.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Cruz, F. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
[CrossRef]

de Miranda, M. H. G.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

Diddams, S. A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Drullinger, R. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Dumke, R.

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Eichenseer, M.

A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002).
[CrossRef]

Elman, V.

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Foreman, S. M.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007).
[CrossRef] [PubMed]

Fortier, T. M.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Gill, P.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

S. A. Webster, M. Oxborrow, and P. Gill, "Subhertz-linewidth Nd:YAG laser," Opt. Lett. 29,1497-1499 (2004).
[CrossRef] [PubMed]

Hall, J. L.

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

C. Salomon, D. Hils, and J. L. Hall, "Laser stabilization at the millihertz level," J. Opt. Soc. Am. B 5,1576-1587 (1988).
[CrossRef]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Hänsch, T. W.

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

Hils, D.

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Huang, X.

Hume, D. B.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Itano, W. M.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
[CrossRef]

Jun Ye, S. M.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

Kolachevsky, N.

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Li, T. C.

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

Liu, T.

T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, "Characterization of the absolute frequency stability of an individual reference cavity," Opt. Lett. 34,190-192 (2009).
[CrossRef] [PubMed]

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Lorini, L.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Lu, Z. H.

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Ludlow, A. D.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, "Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10?15," Opt. Lett. 32,641-643 (2007).
[CrossRef] [PubMed]

Martin, M. J.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

Matveev, A.

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

Millo, J.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Nazarova, T.

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83,531-536 (2006).
[CrossRef]

Nevsky, A. Yu.

A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002).
[CrossRef]

Newbury, N. R.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Notcutt, M.

Oskay, W. H.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Oxborrow, M.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

S. A. Webster, M. Oxborrow, and P. Gill, "Subhertz-linewidth Nd:YAG laser," Opt. Lett. 29,1497-1499 (2004).
[CrossRef] [PubMed]

Peik, E.

T. Schneider, E. Peik, and C. Tamm, "Sub-Hertz optical frequency comparisons between two trapped 171Yb+ ions," Phys. Rev. Lett. 94,230801 (2005).
[CrossRef] [PubMed]

Premoli, A.

A. Premoli and P. Tavella, "A revisited three-cornered hat method for estimating frequency standard instability," IEEE Trans. Instrum. Meas. 42,7-13 (1993).
[CrossRef]

Pugla, S.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

Riehle, F.

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83,531-536 (2006).
[CrossRef]

Rosenband, T.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Salomon, C.

Schmidt, P. O.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Schneider, T.

T. Schneider, E. Peik, and C. Tamm, "Sub-Hertz optical frequency comparisons between two trapped 171Yb+ ions," Phys. Rev. Lett. 94,230801 (2005).
[CrossRef] [PubMed]

Stalnaker, J. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Stejskal, A.

T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, "Characterization of the absolute frequency stability of an individual reference cavity," Opt. Lett. 34,190-192 (2009).
[CrossRef] [PubMed]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

Sterr, U.

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83,531-536 (2006).
[CrossRef]

Swann, W. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Tamm, C.

T. Schneider, E. Peik, and C. Tamm, "Sub-Hertz optical frequency comparisons between two trapped 171Yb+ ions," Phys. Rev. Lett. 94,230801 (2005).
[CrossRef] [PubMed]

Tavella, P.

A. Premoli and P. Tavella, "A revisited three-cornered hat method for estimating frequency standard instability," IEEE Trans. Instrum. Meas. 42,7-13 (1993).
[CrossRef]

Thomsen, J. W.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

Udem, Th.

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

von Zanthier, J.

A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002).
[CrossRef]

Walther, H.

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002).
[CrossRef]

Wang, L. J.

T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, "Characterization of the absolute frequency stability of an individual reference cavity," Opt. Lett. 34,190-192 (2009).
[CrossRef] [PubMed]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

Wang, Y. H.

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

Webster, S. A.

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

S. A. Webster, M. Oxborrow, and P. Gill, "Subhertz-linewidth Nd:YAG laser," Opt. Lett. 29,1497-1499 (2004).
[CrossRef] [PubMed]

Wineland, D. J.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

Yang, T.

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

Ye, J.

Young, B. C.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
[CrossRef]

Zang, E. J.

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

Zanon-Willette, T.

Zelevinsky, T.

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
[CrossRef] [PubMed]

Zhang, J.

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Zhao, Y. N.

T. Liu, Y. N. Zhao, V. Elman, A. Stejskal, and L. J. Wang, "Characterization of the absolute frequency stability of an individual reference cavity," Opt. Lett. 34,190-192 (2009).
[CrossRef] [PubMed]

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

Appl. Phys. B

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31,97-105 (1983).
[CrossRef]

T. Liu, Y. H. Wang, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Narrow linewidth light source for an ultraviolet optical frequency standard," Appl. Phys. B 87,227-232 (2007).
[CrossRef]

T. Nazarova, F. Riehle, and U. Sterr, "Vibration-insensitive reference cavity for an ultra-narrow-linewidth laser," Appl. Phys. B 83,531-536 (2006).
[CrossRef]

IEEE Trans. Instrum. Meas.

A. Premoli and P. Tavella, "A revisited three-cornered hat method for estimating frequency standard instability," IEEE Trans. Instrum. Meas. 42,7-13 (1993).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

Y. H. Wang, T. Liu, R. Dumke, A. Stejskal, Y. N. Zhao, J. Zhang, Z. H. Lu, L. J. Wang, Th. Becker, and H. Walther, "Improved absolute frequency measurement of the 115In+ 5s2 1S0-5s5p 3P0 narrowline transition: progress towards an optical frequency standard," Laser Phys. 17,1017-1024 (2007).
[CrossRef]

Opt. Commun.

A. Yu. Nevsky, M. Eichenseer, J. von Zanthier, and H. Walther, "A Nd:YAG Laser with short-term frequency stability at the Hertz-level," Opt. Commun. 210,91-100 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev. A

L. S. Chen, J. L. Hall, J. Ye, T. Yang, E. J. Zang, and T. C. Li, "Vibration-induced elastic deformation of Fabry-Perot cavities," Phys. Rev. A 74,053801 (2006).
[CrossRef]

J. Alnis, A. Matveev, N. Kolachevsky, Th. Udem, and T. W. Hänsch, "Subhertz linewidth diode lasers by stabilization to vibrationally and thermally compensated ultralow-expansion glass Fabry-P’erot cavities," Phys. Rev. A 77,053809 (2008).
[CrossRef]

S. A. Webster, M. Oxborrow, S. Pugla, J. Millo, and P. Gill, "Thermal-noise-limited optical cavity," Phys. Rev. A 77,033847 (2008).
[CrossRef]

Phys. Rev. Lett.

B. C. Young, F. C. Cruz, W. M. Itano, and J. C. Bergquist, "Visible lasers with subhertz linewidths," Phys. Rev. Lett. 82,3799-3802 (1999).
[CrossRef]

T. Schneider, E. Peik, and C. Tamm, "Sub-Hertz optical frequency comparisons between two trapped 171Yb+ ions," Phys. Rev. Lett. 94,230801 (2005).
[CrossRef] [PubMed]

Science

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, "Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place," Science 319,1808-1812 (2008).
[CrossRef] [PubMed]

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, Jun Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, "Sr lattice clock at 1×10?16 fractional uncertainty by remote optical evaluation with a Ca clock," Science 319,1805-1808 (2008).
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Figures (10)

Fig. 1.
Fig. 1.

(a) Three-dimensional view of the cavity 3 geometry. L is the cavity length, excluding the two mirrors, position D is along the cavity axis. (b) Picture of Cavity 3.

Fig. 2.
Fig. 2.

Simulated result of dL/dg at different supporting point of D, where dL presents the change in cavity length, dg stands for a small change in the acceleration g. The black square curve and the red circle curve indicate the values of dL/dg with the different displacements at D when dg = 0.0981 m/s2, and dg = 0.981 m/s2, respectively.

Fig. 3.
Fig. 3.

The cavity ring down signal of cavity 3, shown with the black curve. The red curve is the exponential fitting of the decay time.

Fig. 4.
Fig. 4.

(a) Schematic of the MISER locking to cavity 1. (b) Schematic of the MISER locking to cavity 2. (c) Schematic of the diode laser locking to a pre-cavity in lab 2. (d) Schematic of the diode laser locking to cavity 3. EOM, electro-optic modulator; AOM, acousto-optic modulator; λ/2, half waveplate; λ/4, quarter waveplate; PBS, polarization beam splitter; PD, photodiode; FC, fiber coupler; M, mirror; L, lens.

Fig. 5.
Fig. 5.

Schematic of the fiber noise compensation. Output 1, the laser beam stabilized to cavity 1; AOM, acousto-optic modulator; λ/2, half waveplate; λ/4, quarter waveplate; PBS, polarization beam splitter; P, polarizer; PD, photodiode; FC, fiber coupler; 10%R, 10% reflection beam splitter.

Fig. 6.
Fig. 6.

Optical beat signals between the three cavities. For the SR785 FFT spectrum analyzer, a frequency resolution of 0.5 Hz is used by choosing a 400 Hz span with a 2 s acquisition time. For each beat note, 10 scans are recorded and averaged. The red curves are Lorentzian fit results.

Fig. 7.
Fig. 7.

The Allan deviations of the beat frequencies between three reference cavities. The linear frequency drifts of approximately 1 Hz/s are removed during data processing.

Fig. 8.
Fig. 8.

(a) The Allan deviations of the individual reference cavities calculated from the classical three-cornered-hat measurement. (b) The Allan deviations of the individual reference cavities with the correlations removed according to [20].

Fig. 9.
Fig. 9.

Estimated correlations for the three reference cavity pairs.

Fig. 10.
Fig. 10.

Comparison between the fourth-order correlation and three times the Allan variance squared of one set of the three beat signals (between cavity 1 and 3) confirms the Gaussian nature of the signals.

Tables (1)

Tables Icon

Table 1. Summary of the cavities specifications.

Equations (30)

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fremote=f0+fAOM+fnoise+fmod,
flocal=f0+2(fAOM+fnoise+fmod).
σi2(τ)=[σij2(τ)+σik2(τ)σik2(τ)]/2.
S=[S11S12S21S22].
[S11S12S21S22]=[r11+r332r13r12+r33r13r23r12+r33r13r23r22+r33r23],
r11=s11r33+2r13 ,
r12=s12r33+r13+r23,
r22=s22r33+2r23.
R=[r11r12r13r12r22r23r13r23r33].
c1=3Ss12(s11s12)(s22s12),
c2=2.25S2+2(s11+s12+s12)c1/(3S),
c3=3S3/2 (s11+s22) +c1/3,
c4=S[1.5S+(s11+s22s12)(s11+s22+s12)],
c5=S3/2 (s11+s22) ,
c6=S2/4 ,
c1f+c2f2+c3f3+c4f4+c5f5+c6f6=0
b0=Ss122+[s122(s11+s22)]f+Ss122f2,
b1=Ss12(2s122+3S/2)fS(s11+s22)f2Sf3/2,
b2=S+2(s11+s22s12)f+3Sf2 .
r33=b1/b2,
r13=r33a10(a02a11)a20a02a112,
r23=r33a10(a20a11)a20a02a112,
a20=2S+fs22,
a02=2S+fs11,
a11=Sfs12,
a10=a01=S (2r33+s12) .
σx=rxx.
<Δx1Δx2Δx3Δx4>=<Δx1Δx2><Δx3Δx4>+<Δx1Δx3><Δx2Δx4>
+<Δx1Δx4><Δx2Δx3>.
ξij4<(yiyj)4>=3<(yiyj)2>2=3(σij2)2,

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